Minors of plane digraphs

TL;DR

This paper introduces a width measure for plane digraphs based on semi-strong minors and demonstrates that excluding a fixed semi-strong minor results in bounded width, unlike the general case.

Contribution

It defines a new width measure for plane digraphs and proves boundedness when excluding a specific semi-strong minor.

Findings

Plane digraphs excluding a fixed semi-strong minor have bounded width.

Plane digraphs in general have unbounded width.

A new branch-composition based width measure is introduced.

Abstract

A digraph $H$ is a ``semi-strong minor'' of another, $G$, if a subdivision of $H$ can be obtained from a subdigraph of $G$ by contracting strongly-connected subdigraphs to single vertices. We will define a width measure of ``plane'' digraphs (that is, drawn in the plane) based on a kind of branch-composition, and show that for every plane digraph $H$, all plane digraphs not containing $H$ as a semi-strong minor have bounded width, while plane digraphs in general have unbounded width.